I could fit it all in one step by using MCMC for example (my model is more complicated than that, so I’ll have to do it by MCMC). However, I could fit the first regression only using MCMC because those estimates are hard to obtain and perform the second regression using least squares or a separate MCMC.

So there’s an ‘one step’ inference based on doing it all at the same time and a ‘two step’ inference by fitting one and using the estimates on the further steps. What is gained or lost between both? Is anything done in this question?

My response:

Rather than answering your particular question, I’ll give you my generic answer, which is to simulate fake data from your model, then fit your model both ways and see how the results differ. Repeat the simulation a few thousand times and you can make all the statistical comparisons you like.